Chapter 2. Measurements in Chemistry
2.1 Measurement Systems
Measuring system is an integral part of any commerce or science to
agree on the quantities of in their business and practices. In England a
measuring system for the trading volumes was not properly standardized
until the 13th century. For example, until gallon was made the standardized
measure of volume consolidating different types of gallons (ale, wine and
corn).
In France and in 1799, a decimal system called metric system using
centimeter, gram, and second (CGS system) for length, mass and time,
respectively. All units are compared to a standard measure: meter was
defined as being one ten-millionth part of a quarter of the earth's
circumference and 100 centimeters equals a meter.
Prefixes used in abbreviating measurements
Prefixes allow simplifying the numbers with many zeros before and after
the decimal place.
Convert measurements to a unit that replaces the power of ten by a
prefix:
6.80 x 10-9 m (6.08 nm) 7.14 x 10-6 s (7.14 m) 2.88 x 10-3 g (2.88 mg)
2.54 x 10-2 m (2.54 cm) 4.56 x 103 g (6.08 kg) 7.14 x 106 s (7.14 Ms)
2.2 Metric System Units-SI system
Later a comprehensive Le Systeme international d'Unites (SI unit
system) was created in 1960 and has been officially adopted by nearly all
countries. SI units are based prefixes and upon 7 principal units, 1 in each
of 7 different categories measurements.
Category
Length
Mass
Time
Electric current
Name
metre
kilogram
second
ampere
Abbrev.
m
kg
s
A
Temperature
kelvin
K
Amount of
mole
Luminous intensity candela
Mol
cd
2.3 Exact and Inexact Numbers
Exact Measurements For certain types of number associated with
conversion factors are considered “exact.” For example, there are exactly
16 ounces in one pound.
places
or
significant
The number 16 would have as many decimal
figures
as
needed.
So
one
pound
has
16.000000000000.... ounces. If you see any numbers you use in a
calculation comes from a definition you could assume they are exact
numbers.
Inxxact Measurements In most of measurements the number associated
with them are considered “inexact.” For example, if you measure the mass
of a certain object on a balance and found that it has a mass of 25.0125 g,
25.013, 25.01 g 25.0 g or 25 g with a decimal place rounded off to a place
at right depending on the decreasing accuracy of the balances used. If you
use a number in a calculation which comes from an experimental
measurement you could safely assume they are inexact numbers.
Which types of numbers are considered “exact?” Below are the general
rules. Metric to metric system conversion factors are exact
1. e.g. 1 m = 100 cm or 1 m/100 cm (In this conversion, 1 and 100 are
both exact.)
Conversions between English and Metric system are generally NOT
exact. Exceptions will be pointed out to you.
2. e.g. 1 in = 2.54 cm exactly (1 and 2.54 are both exact.)
3. e.g. 454 g = 1 lb or 454 g/1 lb (454 has 3 sig. fig., but 1 is exact.)
An example of an extensive property is
a. color. b. freezing point. c. length. d. density.
a
b
c
d
e
2.4 Uncertainty in Measurement and Significant Figures
An
inexact
numbers
always
comes
out
of
an
experimental
measurement. They have been rounded off to show the uncertainty of the
measurement. For example, if you measure the mass of a certain object on
a balance and found that it gives a mass of 25.0125 g, 25.013, 25.01 g
25.0 g or 25 g with a decimal place rounded off to a place at right
depending on the decreasing accuracy of the balances used. A measurement
is always written down after considering the instrumental uncertainties. The
uncertainly of a measurement could be conveniently expressed as a
significant figure (SF). The right most digit in 25.0125 g which is 5 at
the 4th decimal place is considered uncertain digit. Describe uncertainty
and significant figures and how they are obtained for a measurement. The
significant figure for this inexact number is obtained counting the other
digits to the left from the uncertain digit. Therefore, 25.0125 g would have 6
significant figures: 25.013 (5 SF) , 25.01 g (4 SF) 25.0 (3 SF) g and 25 g
(2 SF). Lower the SF, higher the uncertainty of the number and less
accurate.
GENERAL RULES FOR FIGURING WHICH NUMBERS are significant
1. ZEROS used to "place" the decimal are NOT significant figures:
0.015 g = 2 SF
2. LEADING ZEROS BEFORE all the digits are NOT significant:
000340 = 3 SF and 0.000216 g = 3 SF
3. TRAILING ZEROS after all the digits are SIGNIFICANT:
1.500 g= 4 SF
4. SANDWICH ZERO WITHIN a number are SIGNIFICANT:
0.0105 g = 3 SF and 10.5 g = 3 SF
0.027 g = 2 SF (LEADING zeros BEFORE all the digits are NOT significant)
2.600 m = 4 SF (TRAILING zeros after all the digits are SIGNIFICANT)
210.05 s = 5 SF (SANDWICH zeros WITHIN a number are SIGNIFICANT)
0.0306 Kcal = 3 SF (LEADING zeros BEFORE as well as SANDWICH zeros
WITHIN are SIGNIFICANT)
How many significant figures are in the following numbers?
a) 0.0945 (3 SF) b) 83.22 (4 SF)
c) 106 (3 SF) d) 0.000130 (3 SF)
Deduce the number of significant figures contained in the following:
a) 16.0 cm (3 SF) b) 0.0063 m (2 SF) c) 100 km (3 SF)
d) 2.9374 g (5 SF) e) 1.07 lb/in2 (3SF)
How many significant figures are in the following measurements?
a) 25.9000g (6 SF) b) 102 cm (3 SF) c) 0.002 m (1 SF) d) 2001 kg (5
SF) e) 0.0605 s (3 SF) f) 21.2 m (3 SF) g) 0.023 kg (2 SF) h) 46.94 cm
(4 SF)
i) 453.59 g (5 SF)
j) 1.6030 km (5 SF)
Uncertainty, Error, Accuracy, and Precision of measurements
Uncertainty is expressed in terms of the rounding off to a significant figure.
e.g. 25.013 ( 5 SF) , 25.01 g ( 4 SF) 25.0 (3 SF) g and 25 g( 2 SF). Note
that greater the number of significant figures, the greater the precision.
Precision versus Accuracy:
Precision = How close measurements agree: If you take more reading of
the same measurement how close they are.
e.g. 25.0125 g, 25.0124 g and 25.0126 g are more precise than 25.0225 g,
25.10127 g and 25.0326.
Accuracy = how close measurement is to the true value: something could
be precise but inaccurate if the value is off by calibration error.
e.g. Measurements 25.0125 g, 25.0124 g and 25.0126 and the true value
25.0125 g are both precise and accurate.
e.g. Measurements 25.0125 g, 25.0124 g and 25.0126 and the true value
25.0125 g are both precise and accurate.
e.g. 25.0225 g, 25.10127 g and 25.0326 g and the true value 26.0125 g are
neither precise nor accurate.
2.5 Significant Figures and Mathematical Operations
Most of the experiments involve calculation of a answer (derived
quantity) from basic measurements with various units to a complex quantity
with derived units. A simple example would be calculation of velocity of an
object (car) travelling certain distance 356.5 miles in a given time in 4.8
hours. Question is how we obtain a velocity with correct uncertainties (SF)
corresponding to uncertainties of distance and the time. Some of derived
quantities involve additions/subtractions and/or multiplication/division.
General rules for significant figure in addition/subtraction:
1. When adding or subtracting numbers, all numbers must have the same
units.
2. The answer can have no more decimals than the measurement with the
fewest DECIMALS.
254 mL
- 54.1 mL
208.1 mL (4 SF)
208 mL (3 SF)
125.4
g
2.54 g
======
127.94 g
127.9 g
(4 SF)
Rounding Off Numbers
1. "extra" digit is LESS than 5-drop it.
2. "extra" digit is MORE than 5-ADD 1.
3. "extra" digit is 5 “Odd rule“
e.g. 2.535 is rounded as “2.54
4. "extra" digit is 5 “Even rule“
e.g. 2.525 is rounded as “2.52”
Significant figures in multiplication/division calculations
1. When multiplications or divisions of numbers, all numbers must have the
same units.
2. The answer can have no more significant figures than the measurement
with the fewest SIGNIFICANT FIGURES.
(231.54 * 43)/433.4 = 22.972358 (231.54 (5 SF) 43(2 SF) 433.4 (4 SF)
= 22.972358 = 21 (2 SF)
= 21 (2 SF)
Calculate 3.21 cm x 15.091 cm =18.301 cm = Ans. 18.3 cm (3 SF)
Calculate 3.82 x 1.1 x 2.003 = 8.416606 = Ans. 8.4 (2 SF)
Calculate 13.87 ÷ 1.23 = 11.27642276 = Ans. 11.3 (3 SF)
Calculate 0.095 ÷ 1.427 = 0.066573231 = Ans. 0.067 (2 SF)
In a long calculation involving mixed operations, carry as many digits as possible
through the entire set of calculations and then round the final result appropriately.
For example,
(5.00 / 1.235) + 3.000 + (6.35 / 4.0)
=4.04858... + 3.000 + 1.5875=8.630829... 5=8.6
2.6 Scientific Notation
Scientific notation uses power-of-10 to express an extremely large or
small numbers. Scientific notation has a regular number with correct
significant figure with a value between 1 to 10, and a power of 10 by which
the regular number is multiplied. E.g. 0.067 is converted to scientific
notation: 6.7 x 10-2
The table shows several examples of numbers written in standard
decimal notation (left-hand column) and in scientific notation (right-hand
column).
Number in decimal form with
significant digits color
Scientific notation with correct
significant digits
1,222,000.00
1.222 x 10
6
34,500.00
3.450 x 10
4
0.00003450000
3.45 x 10
-0.0000000165
-1.65 x 10
-5
-8
Scientific notation makes it easy to multiply and divide gigantic and/or
minuscule numbers. To obtain the product of these two numbers (the
coefficients) are multiplied, and the powers of 10 are added. This produces
the following result:
2.56 x 1067 x -8.333 x 10-54 = (2.56 x -8.333)(1067 x10-54) =
(-21.33248)(1067-54) = (-21.3)(1013)
= (-2.13)(1014) = -2.13 x 1014
Now consider the quotient of the two numbers multiplied in the previous
example:
(3.46 x 1057 ) / (9.431 x 10-75 )
To obtain the quotient, the coefficients are divided, and the powers of 10 are
subtracted. This gives the following:
= ((3.46 / (9.431)) x (1057/10-75)) = ((3.46 / (9.431)) x (1057x10-(-75))
= (3.46 /9.431)) x 10 57+75 = (3.46 /9.431)) x 10 57+75
= 0.366875199 x 10137 = (3.66875199 x 101 ) x 10137
= (3.67) x ( (101 ) x 10137) = (3.67 x (10138) = 3.67 x 10138
2.7 Conversion Factors and Dimensional Analysis
Conversion Factors are used to convert a measurement to another with
different units. They are used for the length, mass, area, volume,
temperature, energy, force and time conversions as listed below:
Conversion Factors
Length
Mass
Area
Volume
1 ft =12 in
1 yd = 3 ft
1 mi 5280 ft =
1mi = 1.609km
1 in =2.54 cm
1 m =3.281 ft
1 lb = 16 oz
1 ton = 2000 lbs
1 lb = 453.59 g
1
1
1
1
1
1
1
1
Energy
1 cal = 4.18681 J
Btu=1.05506E3 J
1 food cal = 1 kcal
Pressure
1 atm = 760 torr
1 atm = 1.01325E5 Pa
1 mmHg = 1 torr
1 mmHg = 1.333E2 Pa
1 Psi = 6.89476E3 Pa
acre =4.048x103 m2
acre =4840 yd2
mile2=2.589x106 m2
mile2= 640 acres
gal = 4 qt
qt = 2 pt
gal = 3.785 L
L = 103 mL
1 mL = 1 cm3
Time
60 min = 1 hr
24 hr = 1 day
365.25 days = 1 year
Temperature
? K = (x) °C +273.15)
? °C = (x) K - 273.15
? °C = (5/9) ((x)°F -32)
? °F = (9/5)(x)°C +32
Simple unit conversations using factor label method
Dimensional Analysis (also called Factor-Label Method or the Unit
Factor Method) is a problem-solving method that uses conversion factors to
convert unit to get the answer with correct units.




First write the measurement need to be converted.
Select the conversion factors from conversion tables.
Line up conversion factors so the units of the desired answers are
obtained.
Unit of bottom (denominator) must cancel when factor is multiplied by
given number.
Length
How many meters are in a 4 cm?
Conversion factor: 1 cm = 10-2 m or
1 cm
1 cm = 10-2 m or
Align the conversion factor to a cancel cm
10-2 m
4 cm
10-2 m
1 cm
= 4 x 10-2 m
How many inches are in 1 meter? Given the conversion factors 1 inch =
2.54 cm and 1 meter = 100 cm
1.00
m
100 cm
1m
1 in
2.54 cm
= 39.37008 m
= 39.3 m
or
1.00
m
x
100 cm
1m
x
1 in
2.54 cm
= 39.37008 m
= 39.3 m
Note digits in blue are exact and was not considered for SF
Convert
2.4 meters to centimeters (Ans: 240 cm or 2.4 x 102 cm)
Mass
How many grams (g) in 150 pounds (lb) given the equalities
1 lb = 0.454 kg and 1 kg = 1000 g?
150 lb
0.454 kg
1000 g
= 68100 g
1 lb
1 kg
Scientific notation allows to show SF clearly.
= 6.81x 104 g
Convert 65.5 centigrams to milligrams (Ans: 655 mg)
Convert 5 liters to cubic decimeters (Ans: 5 dm3)
The density of a substance is 2.7 g/cm3. What is the density of the
substance in kilograms per liter? (Ans: 2.7 kg/L)
A car is traveling 65 miles per hour. How many feet does the car travel
in one second? (Ans: 95 ft/sec)
How many basketballs can be carried by 8 buses?
Given 1 bus = 12 cars, 3 cars = 1 truck, 1000 basketballs = 1 truck
(Ans: 32 000 basketball)
Area units:
How many cm2 are in a m2 (base unit) of area?
Square each number and unit in the conversion factor
1 cm = 10-2 m ; (1 cm)2 = (10-2 m)2 ; (1)2 cm2 = (10-2)2 m2
1 cm2 = 10-4 m2
Volume unit:
How many cm3 are in m3 (base unit) of volume.
1 cm = 10-2 m
Cube each number and unit in the conversion factor
1 cm = 10-2 m; (1 cm) 3 = (10-2 m)3; (1)3 cm3 = (10-2) 3 m3
1 cm3 = 10-6 m3
How many m3 are in m3 (base unit) of volume?
1 m = 10-6 m (Ans: 1 m3 = 10-18 m3)
How many m3 (base unit) of volume are in cm3.
1 m = 106 m (Ans: 1 m3 = 1018 m3)
Chemistry at a Glance: Practice unit conversion Factors using factor
labeled method.
2.8 Density
Density is one of the physical characteristics of a substance that help to
identify the substance. Density (d), is defined as mass per unit volume.
Density is calculated by dividing the mass of an object by its volume. This is
shown in equation form, as follows:
Density = mass ÷ volume
We can calculate the density of a solid, liquid, or gas.
Note the
difference in units in the formulas of the density of a solid and liquid to the
gas. The unit of mass is grams, g. The unit of volume of sold is cubic
centimeters is cm3, liquids milliliters is mL, and for gases is liters L or
cubic meters m3.
Solids: d = grams (g) ÷ cubic centimeters (cm3) = g/cm3
Lliquids: d = grams (g) ÷ milliliters (mL) = g/mL
Gases: d = grams (g) ÷ milliliters (L or m3) = g/L or g/m3
A student determines that a piece of an unknown solid material has
a mass of 5.854 g and a volume of 7.57 cm3. What is the density of
the material, rounded to the correct number of significant digits?
Calculation using a formula:
First: Write the correct formula at the top of your page, and list the knowns
and the unknowns.
M
D=
= ?
M= 5.854 g, V = 7.57 cm3
V
Second: Substitute the known values in the formula
D=
M
V
=
M= 5.854 g
V = 7.57 cm3
= 0.77336 g/cm3
= 7.73 x 101 g/cm3
Calculation using a factor label method:
5.854 g
1
7.57
cm3
= 0.77336
= 7.73 x 101 g/cm3
Aluminum block weighs 14.2 g and has a density of 2.70 g cm-3.
Calculate the volume of the block.
Calculation using a formula:
First: Write the correct formula at the top of your page, and list the knowns
and the unknowns.
M= 14.2 g , 2.70 g cm-3; or = 2.70 g/1 cm-3 ; V= ?
Second: Substitute the known values in the problem
D=
=
M
V
;
V=
M
D
=
M= 14.2 g
D= 2.70 g/1 cm-3
=
5.2593 g/cm3
5.26 g/cm3
Calculation using a factor label method:
14.2 g
1 cm-3
= 5.2593 cm3
= 5.26
2.70 g
The density of water is one gram per cubic centimeter. What is the
density of water in pounds per liter? (Ans: 0.45 lb/L)
2.9 Temperature Scales and Heat Energy
Temperature Scales
Astronomers and other scientists like to use a temperature scale called
"Kelvin." On a Kelvin thermometer water freezes at 273 degrees and boils
at 373 degrees. Zero degrees Kelvin is called "absolute zero." It is the
lowest possible temperature of matter.
Temperature Conversions
?
?
?
?
K = (x) °C + 273.15)
°C = (x) K - 273.15
°C = (5/9) ((x)°F -32)
°F = (9/5)(x)°C +32
To convert degrees Celsius (°C) to Kelvin (K):
oC
-->K ;
? K = (x) °C -273.15)
K = C + 273.15
Simply add 273. (Example, 0 (x) C° in K = ?=)
? K = (0) °C +273.15) = 273.15 K
To convert Kelvin (K) to degrees Celsius (°C):
? °C = (x) K - 273.15
Simply subtract 273 degrees. (Example, 273 K = 0 deg. C)
To convert degrees Celsius (°C) to Fahrenheit (°F)
? °F = (9/5)(x)°C +32
begin by multiplying the degrees Celsius (°C) temperature by 9, then divide
the answer by 5. Finally add 32.
To convert degrees Fahrenheit (°F) to Celsius (°C):
? °C = (5/9) ((x)°F -32)
Begin by subtracting 32 from Fahrenheit (°F) temperature, then multiply by
5 and divide the answer by 9.
Human body temperature is 98.6 oF. Convert this temperature to
a)
oC
and b) K scale.
a) ? °C = (5/9) ((x)°F -32) = (5/9) ((98.6 )°F -32) = 5/9 (66.6) = 37.0 °C
b)
? K = (x) °C + 273.15) ; 98.6 °F = 37.0 °C from a.
? K = (37.0) °C + 273.15) = 310.2 K = 310. K = 3.10 x 102 K
Describe heat energy and how they are measured in calories, dietary
calories and joules.
Chemical Connections: Describe Body Density and Percent Body Fat
and Normal Human Body Temperature
Describe elements